New version of my math book (“Equivalent filters and rebase of filters” section)
I have added to my book, section “Equivalent filters and rebase of filters” some new results. Particularly I added “Rebase of unfixed filters” subsection. It remains to research the properties of the lattice of unfixed filters.
Equivalence filters and the lattice of filters on the poset of small sets
Equivalence of filters can be described in terms of the lattice of filters on the poset of small sets. See new subsection “Embedding into the lattice of filters on small” in my book.
Poset of unfixed filters
I defined partial order on the set of unfixed filters and researched basic properties of this poset. Also some minor results about equivalence of filters. See my book.
Rebase of filters and equivalent filters
I added a few new propositions into “Equivalent filters and rebase of filters” section (now located in the chapter “Filters and filtrators”) of my book. I also define “unfixed filters” as the equivalence classes of (small) filters on sets. This is a step forward to also define “unfixed funcoids” and “unfixed reloids”.
A step forward to solve an open problem
I am attempting to find the value of the node “other” in a diagram currently located at this file, chapter “Extending Galois connections between funcoids and reloids”. By definition $latex \mathrm{other} = \Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}}$. A few minutes ago I’ve proved $latex (\Phi_{\ast}(\mathsf{RLD})_{\mathrm{out}})\bot = \Omega^{\mathsf{FCD}}$, that is found the value of the function “other” at $latex \bot$. It is […]
New short chapter
I’ve added a new short chapter “Generalized Cofinite Filters” to my book.
My math book updated
I have updated my math book with new (easy but) general theorem similar to this (but in other notation): Theorem If $latex \mathfrak{Z}$ is an ideal base, then the set of filters on $latex \mathfrak{Z}$ is a join-semilattice and the binary join of filters is described by the formula $latex \mathcal{A}\sqcup\mathcal{B} = \mathcal{A}\cap\mathcal{B}$. I have updated some […]
My math book updated
I updated my math research book to use “weakly down-aligned” and “weakly up-aligned” instead of “down-aligned” and “up-aligned” (see the book for the definitions) where appropriate to make theorems slightly more general. During this I also corrected an error. (One theorem referred to complement of a lattice element without stating that the lattice is boolean.) Well, maybe I […]
A counterexample to my recent conjecture
After proposing this conjecture I quickly found a counterexample: $latex S = \left\{ (- a ; a) \mid a \in \mathbb{R}, 0 < a < 1 \right\}$, $latex f$ is the usual Kuratowski closure for $latex \mathbb{R}$.
New conjecture about funcoids
Conjecture $latex \langle f \rangle \bigsqcup S = \bigsqcup_{\mathcal{X} \in S} \langle f \rangle \mathcal{X}$ if $latex S$ is a totally ordered (generalize for a filter base) set of filters (or at least set of sets).